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%I #22 Mar 18 2017 16:13:46
%S 1,720,3110940,24046189440,315031400802720,6736218287430460752,
%T 226885231700215713535680,11649337108041078980732943360,
%U 885282776210120715086715619724160,96986285294151066094112970262797953280
%N Number of n X n (0,1) matrices containing exactly five 1's in each row and in each column.
%C Also number of ways to arrange 5n rooks on an n X n chessboard, with no more than 5 rooks in each row and column. - _Vaclav Kotesovec_, Aug 04 2013
%C Generally (Canfield + McKay, 2004), a(n) ~ exp(-1/2)*binomial(n,s)^(2*n) / binomial(n^2,s*n), or a(n) ~ sqrt(2*Pi)*exp(-n*s-1/2*(s-1)^2)*(n*s)^(n*s+1/2)*(s!)^(-2*n). - _Vaclav Kotesovec_, Aug 04 2013
%D B. D. McKay, Applications of a technique for labeled enumeration, Congressus Numerantium, 40 (1983) 207-221.
%H Vaclav Kotesovec, <a href="/A075754/b075754.txt">Table of n, a(n) for n = 5..61</a>, (computed with program by Doron Zeilberger, see link below)
%H B. D. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/data/semiregular.html">0-1 matrices with constant row and column sums</a>
%H E. R. Canfield and B. D. McKay, <a href="http://www.ams.org/mathscinet-getitem?mr=2156683">Asymptotic enumeration of dense 0-1 matrices with equal row and column sums</a>.
%H Shalosh B. Ekhad and Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/bipartite.html">In How Many Ways Can n (Straight) Men and n (Straight) Women Get Married, if Each Person Has Exactly k Spouses</a>, Maple package Bipartite.
%H M. L. Stein and P. R. Stein, <a href="/A001496/a001496.pdf">Enumeration of Stochastic Matrices with Integer Elements</a>, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>
%F From _Vaclav Kotesovec_, Aug 04 2013: (Start)
%F a(n) ~ exp(-1/2)*binomial(n,5)^(2*n) / binomial(n^2,5*n), (Canfield + McKay, 2004)
%F a(n) ~ sqrt(Pi)*2^(1/2-6*n)*5^(3*n+1/2) *9^(-n)*exp(-5*n-8)*n^(5*n+1/2)
%F (End)
%Y Column 5 of A008300.
%K nonn
%O 5,2
%A Michel Buffet (buffet(AT)engref.fr), Oct 08 2002
%E More terms from _Brendan McKay_, Jan 08 2005
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